(a) Let $\mathcal C$ be the triangle with vertices at $\{0,1,i\}$ oriented counterclockwise. Calculate $$\int_{\mathcal C}|z|^2\,dz.$$ (b) Evaluate $$\int_{\mathcal C}z^3e^{-z^4}\,dz$$ along the path $$\mathcal C=\left\{\sin t^2-i\frac{2t^2}\pi:0\le t\le\sqrt{\pi/2}\right\}.$$ (c) Evaluate $$\oint_{|z|=\pi}\frac{\sin z}{z^2(z-\pi/2)}\,dz.$$
I'm not sure if I need to integrate over the $i\to0$ and $0\to1$ line segments as well.
You have only integrated over the line segment $ 1 \to i$. You have to integrate over the the line segments $i→0$ and $0→1$ as well.
Then you have to add the three resulting integrals.